Is That a Stuffed Hamster?

How realistic is it that a writer might forge a career relationship with one
editor?

It does happen, and it's a happy thing for everyone when it does. But it's not very frequent.

Now that I’m about to start sending out queries, I want to target those editors who are looking for manuscripts in the varied genres I write in. (Material suitable for their list of course.) Am I dropping the ball here by thinking this way? Should I concentrate solely on the sale of each individual manuscript and not consider this?

Yes. Trying to get your manuscript published is hard enough without trying to find and only submit to editors who might take any genre you ever feel like writing. Find someone to publish something of yours, and count yourself lucky. If you happen to find someone with whom to build a continuing relationship, count yourself very lucky indeed.

When you take on a writer, do you look ahead for future projects from him or
her, or do you just care specifically about the current one?

I want to be sure you understand something—the writers who do have a continuing relationship with a particular editor have it for these reasons; and all of these reasons, not just one or two of them:

1. Luck (and perhaps some networking). If an editor and an author who make a really great team happen to find each other, there's always some serendipity about it. Erect a shrine to the goddess Bibliomachia and hope. (If you want to make offerings, she prefers burned manuscript pages and broken pencils.)

2. Professionalism and similar working styles. Those editors you see at conferences and hear about in the writing community? 99.9% of them are very nice people. Those authors out there hoping to get published? 99.9% of them are very nice people. This does not matter, because being a nice person and being good to work with are about as related as hamsters and salad forks.
You don't care if your editor is the kind of person who steals lawn gnomes and dribbles spaghetti sauce into library books, as long as she helps you make your book better and gets back to you promptly when you email. Your editor doesn't care if you're the kind of person who taxidermies your dead pets and only washes your hair when it starts to itch, as long as you take feedback and respect your due dates. It doesn't matter how much you and your editor like each other personally; if your working styles drive each other nuts, you'd both be much happier with other people.

3. Success. Here's the real catch. Not one of those happy teams you may have heard about is turning out unsuccessful books. If you sell a manuscript to an editor who you do happen to enjoy working with, wonderful. But if that book bombs, her publisher is going to cast a skeptical eye on any other manuscripts from you that she decides to champion. And she knows this, so she's less likely to champion them. It doesn't matter how great you both are to work with or how blessed by the ferocious goddess of books you both happen to be if you aren't creating books that sell.